Taking roots over high extensions of finite fields.

We present a new algorithm for computing m-th roots over the finite field F-q, where q = p(n), with p a prime, and m any positive integer. In the particular case m = 2, the cost of the new algorithm is an expected O(M(n) log(p) + C(n) log(n)) operations in F-p, where M(n) and C(n) are bounds for the cost of polynomial multiplication and modular polynomial composition. Known results give M(n) = O(n log(n) log log(n)) and C(n) = O(n(1.67)), so our algorithm is subquadratic in n.